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Sullivan PreCalculus
Section 2.1
Functions
Objectives
• Determine Whether a Relation Represents a Function
• Find the Value of a Function
• Find the Domain of a Function
• Form the Sum, Difference, Product and Quotient of Two
Functions
Let X and Y be two nonempty sets of real
numbers. A function from X into Y is a
relation that associates with each element of
X a unique element of Y.
The set X is called the domain of the
function.
For each element x in X, the
corresponding element y in Y is called the
image of x. The set of all images of the
elements of the domain is called the
range of the function.
f
x
y
x
y
x
X
DOMAIN
Y
RANGE
Example: Which of the following relations
are function?
{(1, 1), (2, 4), (3, 9), (-3, 9)}
A Function
{(1, 1), (1, -1), (2, 4), (4, 9)}
Not A Function
Functions are often denoted by letters such as f,
F, g, G, and others. The symbol f(x), read “f of
x” or “f at x”, is the number that results when x is
given and the function f is applied.
Elements of the domain, x, can be though of as
input and the result obtained when the function
is applied can be though of as output.
Restrictions on this input/output machine:
1. It only accepts numbers from the
domain of the function.
2. For each input, there is exactly one
output (which may be repeated for
different inputs).
For a function y = f(x), the variable x is
called the independent variable, because it
can be assigned any of the permissible
numbers from the domain.
The variable y is called the dependent
variable, because its value depends on x.
The independent variable is also called
the argument of the function.
Example: Given the function f ( x)  2x  5
2
Find: f (3)
f (3)  2(3)  5  23
2
f (x) is the number that results when the
number x is applied to the rule for f.
Find: f ( x  h)
f ( x  h )  2( x  h )  5
2
 2( x  2 xh  h )  5
2
2
 2x  4xh  2h  5
2
2
The domain of a function f is the set of real
numbers such that the rule of the function
makes sense.
Domain can also be thought of as the set of
all possible input for the function machine.
Example: Find the domain of the following function:
g ( x )  3x  5x  1
3
Domain: All real numbers
Example: Find the domain of the following function:
4
s( t ) 
t 1
Domain of s is  t | t  1 .
Example: Find the domain of the following function:
h( z )  z  2
z20
z  2
Domain of h is  z| z  2 .
Example: Express the area of a circle as a function of
its radius.
A(r )  r
2
The dependent variable is A and the independent
variable is r.
The domain of the function is {r | r  0}
If f and g are functions, their sum f + g is the
function given by
(f + g)(x) = f(x) + g(x).
The domain of f + g consists of the numbers x that
are in the domain of f and in the domain of g.
If f and g are functions, their difference f - g is
the function given by
(f - g)(x) = f(x) - g(x).
The domain of f - g consists of the numbers x that
are in the domain of f and in the domain of g.
Their product f  g is the function given by
(f  g)(x) = f(x)  g(x)
The domain of f  g consists of the numbers x that
are in the domain of f and in the domain of g.
Their quotient f / g is the function given by
(f / g)(x) = f(x) / g(x), g(x)  0
The domain of f / g consists of the numbers x for
which g(x) 0 that are in the domain of f and in
the domain of g.
Example: Define the functions f and g as follows:
f ( x)  x  3
g ( x)  x 2  16
Find each of the following and determine the
domain of the resulting function.
a.) (f + g)(x) = f(x) + g(x)

x  3  x  16
2
The domain of f consists of all real numbers x for
which x > 3; the domain of g consists of all real
numbers.
The domain of f + g is {x | x > 3}.
b.) (f - g)(x) = f(x) - g(x)


 x  3  x  16

2
x  3  x 2  16
The domain of f - g is {x | x > 3}.
c.) ( f  g )(x)= f(x)g(x)

  x  3 x 2  16

The domain of f  g is {x | x > 3}.
f ( x) 
x3
g ( x )  x  16
2
f ( x)
d.) ( f / g )( x ) 
g( x)
x3
 2
x  16
We must exclude x = - 4 and x = 4 from the
domain since g(x) = 0 when x = 4 or - 4.
The domain of f / g is {x | x > 3, x  4}.